Georgian Mathematical Journal Volume 8 (2001), Number 4, 665–668

ON THE RATIONALITY OF CERTAIN STRATA OF THE LANGE STRATIFICATION OF STABLE VECTOR BUNDLES

ON CURVES

E. BALLICO

Abstract. Let X be a smooth projective curve of genus g ≥ 2 and S(r, d) the moduli scheme of all rank r stable vector bundles of degree d on X. Fix an integer k with 0 < k < r. H. Lange introduced a natural stratification of S(r, d) using the degree of a rank k subbundle of any E ∈ S(r, d) with maximal degree. Every non-dense stratum, say W (k, r − k, a, d − a), has in a natural way a fiber structure h : W (k, r − k, a, d − a) → Pic^{a}(X) × Pic^{b}(X) with h dominant. Here we study the rationality or the unirationality of the generic fiber of h.

2000 Mathematics Subject Classification: 14H60.

Key words and phrases: Stable vector bundles on curves, moduli schemes of vector bundles, rational variety, unirational variety.

1. Introduction

Let X be a smooth complete algebraic curve of genus g ≥ 2 defined over an algebraically closed base field K with char(K) = 0. Fix integers r, d with r ≥ 1 and L ∈ Pic(X). Let S_{L}(r, d) be the moduli scheme of stable rank r vector bundles on X with determinant L and S(r, d) the moduli scheme of all stable rank r vector bundles on X with degree d. It is well-known ([14]) that S(r, d) (resp. S_{L}(r, d)) is smooth, irreducible, of dimension (r^{2} − 1)(g − 1) + g (resp. (r^{2} − 1)(g − 1)) and that S_{L}(r, d) is unirational. The variety S_{L}(r, d) is a fine moduli scheme if and only if (r, d) = 1. P. E. Newstead ([11]) proved in many cases that S_{L}(r, d) is rational. For other cases, see [1]. By [5] S_{L}(r, d) is rational if (r, d) = 1. In [6] H. Lange introduced the following stratification (called the Lange stratification) of the moduli scheme S(r, d), r ≥ 2, depending on the choice of an integer k with 0 < k < r. For any rank r vector bundle E set s_{k}(E) := k(deg(E)) − r(deg(A)), where A is a rank k subsheaf of E with maximal degree. By [9] we have s_{k}(E) ≤ gk(r − k). If E is stable, then s_{k}(E) > 0. By [4], sect. 4, (see [8], Remark 3.14) for any L and a general E ∈ S_{L}(r, d) we have s_{k}(E) = k(r − k)(g − 1) + e, where e is the unique integer with (r − 1)(g − 1) ≤ e ≤ (r − 1)g and e + k(r − k)(g − 1) ≡ kd mod (r). For any integer a set V (k, r − k, a, d − a) := {E ∈ S(r, d) : s_{k}(E) = kd − ra}. This gives a stratification of S(r, d) which will be called the Lange stratification of S(r, d). Here we study the rationality or the unirationality of smaller strata

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of this stratification. Hence (setting b = d − a) we fix integers r, k, a, b with 0 < k < r and a/r < b/(r − k) < a/r + g − 1. By [12], Th. 0.1, there is a non-empty open irreducible subset W (k, r − a, a, b) of V (k, r − k, a, b) such that every E ∈ W (k, r − k, a, b) fits in an exact sequence

0 → H → E → Q → 0  (1)

with H computing s_{k}(E) (i.e. with rank(H) = k, deg(H) = a, rank(Q) = r − k and deg(Q) = b), H and Q stable and such that H is the only rank k subsheaf of E computing s_{k}(E). This means that (up to a scalar) E fits in a unique extension (1). Furthermore, varying E in W (k, r − k, a, a, b), the pairs (H, Q) obtained in this way cover a Zariski dense constructible subset of S(k, a)×S(r− k, b). Conversely, the generic extension of the generic element of S(r − k, b) by the generic element of S(k, a) is the generic element of W (k, r − k, a, b). Hence there is a rational dominant map W (k, r − k, a, b) → Pic^{a}(X) × Pic^{b}(X) ≅ Alb(X) × Alb(X) sending E into (det(H), det(Q)). For any L ∈ Pic^{a}(X) and any M ∈ Pic^{b}(X) set W (k, r − k, a, b, L, M) := {E ∈ W (k, r − k, a, b) : E fits in a unique exact sequence (1) and L ≅ det(H) and M ≅ det(Q) }. We are interested in the rationality or unirationality of the strata W (k, r−k, a, b, L, M). In this paper we prove the following results.

Theorem 1. Fix integers r, k, a, b with 0 < k < r and a/r < b/(r − k) < a/r + g − 1. Then for a general pair (L, M) ∈ Pic^{a}(X) × Pic^{b}(X) the variety W (k, r − k, a, b, L, M) is unirational.

Theorem 2. Fix integers r, k, a, b with 0 < k < r, a/r < b/(r − k) < a/r + g − 1, (k, a) = 1 and (r − k, b) = 1. Then for a general pair (L, M) ∈ Pic^{a}(X) × Pic^{b}(X) the variety W (k, r − k, a, b, L, M) is rational.

Proofs of Theorems 1 and 2

Lemma 1. Fix integers u, v, a and b with u > 0 and v > 0 and take a general pair (L, M) ∈ Pic^{a}(X) × Pic^{b}(X). Then for a general pair (A, B) ∈ S_{L}(u, a) × S_{M}(v, b) we have h^{0}(X, Hom(A, B)) = max{0, bu − av + uv(1 − g)} and h^{1}(X, Hom(A, B)) = max{0, −bu + av + uv(g − 1)}.

Proof. Without the restrictions det(A) ≅ L and det(B) ≅ M, this is a result of A. Hirschowitz (see [2], sect. 4, or [13], Th. 1.2, for a published proof). By semicontinuity and the openness of stability we obtain the result for a general pair (L, M).

Lemma 2. Fix integers u, v, a and b with u > 0, v > 0 and a/u < b/v and take a general pair (L, M) ∈ Pic^{a}(X) × Pic^{b}(X). Then for a general pair (A, B) ∈ S_{L}(u, a) × S_{M}(v, b) the general extension of B by A is stable.

Proof. Without the restrictions det(A) ≅ L and det(B) ≅ M, this is proved in [13] during the proof of [13], Theorems 0.1 and 0.2. By the openness of stability we obtain the result for a general pair (L, M).

Lemma 3. Fix integers u, v, a and b with u > 0, v > 0 and a/u < b/v < a/u + g − 1. Take a general pair (L, M) ∈ Pic^{a}(X) × Pic^{b}(X). Then for a general pair (A, B) ∈ S_{L}(u, a) × S_{M}(v, b) the general extension, E, of B by A is stable, s_{u}(E) = ub − va and A is the only rank u subbundle of E computing s_{u}(E).

Proof. Without the restrictions det(A) ≅ L and det(B) ≅ M, this is [13], Th. 0.1. By the openness of stability and the semicontinuity of the Lange invariant s_{u} we obtain the result for a general pair (L, M).

Now we can prove Theorems 2 and 1.

Proof of Theorem 2. The variety S_{L}(k, a) × S_{M}(r − k, b) is rational by [5], Th. 1.2. Since (k, a) = (r − k, b) = 1, both S_{L}(k, a) and S_{M}(r − k, b) are fine moduli spaces and hence there is a universal family, U, of pairs (A, B) of vector bundles on S_{L}(k, a) × S_{M}(r − k, b). For every (A, B) ∈ S_{L}(k, a) × S_{M}(r − k, b) we have h^{0}(X, Hom(A, B)) = 0 because μ(B) = b/(r − k) > a/k = μ(A) and both A and B are stable. Thus h^{1}(X, Hom(A, B)) = kb − (r − k)a + k(r − k)(g − 1) (Riemann–Roch), i.e. h^{1}(X, Hom(A, B)) does not depend from the choice of the pair (A, B) ∈ S_{L}(k, a) × S_{M}(r − k, b) but only from the integers k, r, a and b. Thus the vector spaces H^{1}(X, Hom(A, B)), (A, B) ∈ S_{L}(k, a) × S_{M}(r − k, b), fit together to form a vector bundle EXT on S_{L}(k, a) × S_{M}(r − k, b): the relative Ext-functor considered in [7]; here we need the existence of U (i.e. the conditions (k, a) = (r − k, b) = 1) for the construction of EXT. Since EXT is a vector bundle over an irreducible rational variety, the total space of EXT is an irreducible rational variety. By [13], Th. 0.1, a non-empty open subset V of EXT corresponds to elements of W (k, r − k, a, b, L, M) and conversely a general element of W (k, r − k, a, b, L, M) corresponds to a general element of EXT. Hence there is a rational dominant map, f, from EXT into W (k, r − k, a, b, L, M). As explained in the introduction, the uniqueness part in [13], Th. 0.1, means that the rational map f induces a generically bijective map from the projective bundle P (EXT)) onto W (k, r − k, a, b, L; M). Since P (EXT)) is rational and char(K) = 0, we conclude.

Proof of Theorem 1. Fix integers x, y with x > 0, P ∈ X and R ∈ Pic^{y}(X). Since S_{R}(x, y) ≅ S_{R(uxP)}(x, y + ux) for every integer u, we will assume y very large, say y > x(2g − 1). By the very construction of S_{R}(x, y), y > x(2g − 1), using Geometric Invariant Theory, there is a smooth variety U_{R}(x, y) with a PGL(N)-action, N = y+x(1−g), without any fixed point and a morphism f_{x,y} : U_{R}(x, y) → S_{R}(x, y) which make S_{R}(x, y) the GIT-quotient of U_{R}(x, y) and such that on U_{R}(x, y) × X there exists a total family of vector bundles on X with R as determinant. We repeat the proof of Theorem 2 using U_{L}(k, a)×U_{M}(r −k, b) instead of S_{L}(k, a) × S_{M}(r − k, b). Since on U_{L}(k, a) × U_{M}(r − k, b) there is a family of pairs of stable vector bundles, we may take a global EXT which is a vector bundle over U_{L}(k, a) × U_{M}(r − k, b) and hence it is irreducible and rational. By [13], Th. 0.1, there are a non-empty open subset V of EXT and a dominant morphism f : V → W (k, r−k, a, b, L, M). Thus W (k, r−k, a, b, L, M) is unirational.

Acknowledgement

The author was partially supported by MURST (Italy).

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(Received 22.02.2001)

Author’s address:

Dept. of Mathematics

University of Trento

38050 Povo (TN)

Italy

E-mail: ballico@science.unitn.it